Limit theorems for random vectors with operator normalizations. I (Q2740444)

Let \(\{S_{n}, n\geq 0\}\) be a sequence of random vectors in \(R^{m},\) and let \(\{A_{n}, n\geq 1\}\) be a sequence of non-random linear operators from \(R^{m}\) to \(R^{d},\) \(m,d\geq 1.\) This paper deals with the general theorem on the strong law of large numbers (LLN) with operator normalizing for arbitrary sequences of random vectors \(S_{n},\) namely, it is proved that \(\|A_{n}S_{n}\|\to 0\) a.s. as \(n\to\infty,\) under some conditions, where \(\|\cdot\|\) is the Euclidian norm. New results on the strong LLN for sequences of sums of independent and orthogonal random vectors and for vector martingales are derived with the help of this general theorem. The main attention is paid to the vector martingales.